De la Viquipèdia, l'enciclopèdia lliure
Tot seguit es presenta una llista de primitives de funcions irracionals. Per consultar una llista completa de primitives de tota mena de funcions adreceu-vos a taula d'integrals
Integrals que contenen l'arrel de (x² +a²)
.
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![{\displaystyle \int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/057de84eaf02efa9ffba665a6094ba3439632d8a)
![{\displaystyle \int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d8606f4f0c607e839021b8b07a06b077ece02f)
![{\displaystyle \int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69530dada3fe7e20bfdd87fba4de51783fc31183)
![{\displaystyle \int xr\;dx={\frac {r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3d78f3e68e641a9b0aab864274113ec4ec57eb)
![{\displaystyle \int xr^{3}\;dx={\frac {r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d723e526aaa761351215a402932ce3cf3dd6b73d)
![{\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6583ead420aa74655131aaf0ed82cbb82e36157)
![{\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697087c789855ff9e55918ad41818251d5b940a7)
![{\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d0c710fe67b89173e00e0f8f877b089029473d)
![{\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c286c93cf9b39345e6c65e118526cf6b0ebba341)
![{\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca1e44ac2d54d893799548feac05e8684a650d4)
![{\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48c6fc659e5a2eb4a86664f238b542283bc1ee2)
![{\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/235dc49fb2ec51fc06ad3c0ed644b5032eed78e5)
![{\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61b0bbd8eb25830f6ff87d1818574f783d0ce222)
![{\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbd4bfbe0c7d57a09ac1901c6522cb7e8327d48)
![{\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982e3258c17a21f19d9bdfa66ccbb2e675a32564)
![{\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37516c175afbb2ba492a368426ca0a68db63935)
![{\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\sinh ^{-1}{\frac {a}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa251ebfb074f5c06ca58c44d521a6b1fc1bea28)
![{\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a99fe55679d157911cd614e9ca510e54d729d42)
![{\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d35f2538f5e4bbde4837e0642fdceb6e151c5062)
![{\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0505fdb3e13b7e0704f61b5285649e041ec8d66)
![{\displaystyle \int {\frac {dx}{r}}=\sinh ^{-1}{\frac {x}{a}}=\ln \left|x+r\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7086008adf22252de463c6ba34333d10d2ec3b2)
![{\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e903aacbfd829184b3eeb9233ce14ac236d1e6b)
![{\displaystyle \int {\frac {x\,dx}{r}}=r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a3156a33834a19e192af67f3a54da6ddbe4ce1)
![{\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8efc8a15f3072555233b2781dc4c398780f904e6)
![{\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\sinh ^{-1}{\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41b6e611d94c33fbbd8d62e7faaf8b7ab8843f8b)
![{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\sinh ^{-1}{\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62b641b02f29a54dfae30532df23f40d76d25a6b)
Integrals que contenen l'arrel de (x² -a²) ![{\displaystyle s={\sqrt {x^{2}-a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c805d83f7cc45cb4a853fc0db6ac23c36bc908e)
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En aquest cas ha de ser
, per a
, consulteu la propera secció.
![{\displaystyle \int xs\;dx={\frac {1}{3}}s^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ee984a361d44c45466b1c0d5bdddc61425ca6c)
![{\displaystyle \int {\frac {s\;dx}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3796397e7bf4a7e7654dd17d20474cf8bdec98)
![{\displaystyle \int {\frac {dx}{s}}=\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0888676523244a1761d956f94a726af0e3617f4d)
Fixeu-vos que
, on s'ha d'agafar el valor positiu del
.
![{\displaystyle \int {\frac {x\;dx}{s}}=s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17011581cfa3d9e53920f319ac77a75013dec77)
![{\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f2035386db8b8331fafbbb54ba1c1cf994fb5d)
![{\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bcfb12b7156194900353c9def5d1e095149ce99)
![{\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db234e0dc43716a52f17f53c9b79e30303aaf18)
![{\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/808ffdc762b49c281071fd22b77938afa3e0cd5a)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819c367c3f01849bfcfa4f310a2e2fc8d4630d2e)
![{\displaystyle \int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25191a1c6fa21c48b6b89f476b96e8c7fa54a541)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755124bfe54faeb5f7029ee29993b42e0335da23)
![{\displaystyle \int {\frac {x^{4}\;dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95143cd1e023e990284ba780dcf4d8d6089ac254)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ba248b7ebdb39ad25a8b8e98996b7b7994162e)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d29db4a76fe21b51b57e5e47e8136b88280fd5)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56fdbdff0c7f9f2437cd5c03b5772496597a6ab)
![{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676ac2308ed60218f4e246884e5783df8e2ebc54)
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![{\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![{\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0c92cdcb44ecfe7179711341a1964ef2a0782f)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ea4b7b2973dd3e2affa09931a8bf41316161f1)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/239ff6c41a3342440c712b9f0c4940e8e6a000d2)
Integrals que contenen l'arrel de (a²-x²)![{\displaystyle u={\sqrt {a^{2}-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95adab5c59d94f2b08b40791068567570b43e9a7)
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![{\displaystyle \int u\;dx={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c38e2826766e3ba4feca78db794b4176f5cb96d)
![{\displaystyle \int xu\;dx=-{\frac {1}{3}}u^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eed0f3cd0c0d48a63c480f4a1e2eb4da5951fa43)
![{\displaystyle \int {\frac {u\;dx}{x}}=u-a\ln \left|{\frac {a+u}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85f145be76d4f31f6e306b127e3e8dd2bf0c30e)
![{\displaystyle \int {\frac {dx}{u}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dacefb8fb4df4a4cd90b7e24706b39ce53a66023)
![{\displaystyle \int {\frac {x^{2}\;dx}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d027d6e3073d768e6d434e22e9e55305b7857cb)
![{\displaystyle \int u\;dx={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\cosh ^{-1}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(per }}|x|\geq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6ab1c0dd296d8213e662ee278620916da73d78)
Integrals que contenen l'arrel de (ax²+bx+c) ![{\displaystyle R={\sqrt {ax^{2}+bx+c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2799cbf609e20e98fdfb5064f181935257d025)
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![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{(per }}a>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55a6d94121e722de4fb9b1e101cdac408d875960)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\sinh ^{-1}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(per }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8b1960fcab6229a0a946482176a6fbed61c6f1)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(per }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0abb312bf4189f511c7f3bb484cfb7987861aac5)
![{\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(per }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left|2ax+b\right|<{\sqrt {b^{2}-4ac}}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32cfe964fd7ca1e46897d995e384a84a1200232e)
![{\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/086934b294e8b53bebe7b53241bad912f4212dee)
![{\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6887eff55e44af7ed031fa1d919d3de3f379a90b)
![{\displaystyle \int {\frac {dx}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {dx}{R^{2n-1}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd19c82abfd6ab01d93cc3f2691059d4b4915c)
![{\displaystyle \int {\frac {x}{R}}\;dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6880a823009f7d8fe7e2579fa228324ef0d15f03)
![{\displaystyle \int {\frac {x}{R^{3}}}\;dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96545813e77cd2cc46ac4034b607a9457df212a0)
![{\displaystyle \int {\frac {x}{R^{2n+1}}}\;dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de6de2ef912d111ee50377af9daaf84b45633498)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {c}}R+bx+2c}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4c2981ae219f385ff1e91f03659b9d15b9a836)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\sinh ^{-1}\left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae837397e05d3194389f794678010cf4b72c74b)
Integrals que contenen l'arrel de (ax+b) ![{\displaystyle S={\sqrt {ax+b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f15909647c04a43592ea6d8f3b4a2d1a060d727c)
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![{\displaystyle \int {\frac {dx}{\sqrt {ax+b}}}\,=\,{\frac {2{\sqrt {ax+b}}}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/925efdb9ab2560fd2149377e096b0b813cb207b2)
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95499b4d680cece94732be56d57e74c57ea51ec9)
![{\displaystyle \int {\frac {\sqrt {ax+b}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ddf991f7c4b497fc370e5bac1c7df2f39022ac)
![{\displaystyle \int {\frac {x^{n}}{\sqrt {ax+b}}}\,dx\;=\;{\frac {2}{a(2n+1)}}\left(x^{n}{\sqrt {ax+b}}-bn\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc428e49a57674b857b5c7373aa652a042179e60)
![{\displaystyle \int x^{n}{\sqrt {ax+b}}\,dx\;=\;{\frac {2}{2n+1}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int x^{n-1}{\sqrt {ax+b}}\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8c3bf8206cec454702eaab4cc9d3bf911983b1)